aim: graph of best fit course: alg. 2 & trig. aim: how do we model real-world data with...

Aim: Graph of Best Fit Course: Alg. 2 & Trig.

Aim: How do we model real-world data with polynomial and other functions?

Do Now: 6 pt. Regents QuestionThe 1999 win-loss statistics for the American League East baseball teams on a particular date is shown in the accompanying chart.

W L

New York 52 34

Boston 49 39

Toronto 47 43

Tampa Bay 39 49

Baltimore 36 51

Find the mean for the number of wins, , and the mean for the number of losses, , and determine if the point ( is a point on the line of best fit. Justify your answer.

WL

,W )L


Model Problem

The average daily amount of waste generated by each person in the United States is given below. This includes all wastes such as industrial wastes, demolition wastes, and sewage.

Yr. 1980 1985 1990 1991 1992 1993 1994 1995 1996

lbs. of waste per

person per day

3.7 3.8 4.5 4.4 4.5 4.5 4.5 4.4 4.3

Create a scatter plot and determine the regression line. Round to nearest

hundredth

y = .05x – 98.69


Correlation

Positive Correlation•y tends to increase as x

increases•slope is positive

No Correlation

Negative Correlation•y tends to decrease as x

increases•slope is negative


Correlation Co-efficient

Data that are linear in nature will have varying degrees of goodness of fit to the lines of fit.

The correlation coefficient r describes the nature of data.

The closer the fit of the data to the line, the closer r gets to + 1 or -1

0 < r < 0.5 positive/weak

0.75 < r < 1 strongly positive

-0.5 < r < 0 moderately

negative


Real World Data & Poly Function Shapes

linear y = ax + b

4

2

4

2

5

quadratic y = ax2 + bx + c

4

2

cubic y = ax3+bx2+ cx+d

-2

-4

quartic y = ax4 + bx3 + cx2 + dx + e

No Direction Change

1 Direction Change

2 Direction Changes

3 Direction Changes


Which Function is Best Fit?

2

-2

-4

Determine the type of polynomial function that could be used to represent the data in each scatter plot.

2

-2

-4

Two direction change: cubic function would be

best fit

One direction change: quadratic function would

be best fit


Functions Modeling Data

Write a polynomial function that models the set of data.

x -1 -.5 0 0.5 1 1.5 2 2.5 3 3.5 4

f(x) -10 -6.4 -5 -5.1 -6 -6.9 -7 -5.6 -2 4.6 15

enter x into L1

enter f(x) into L2

View Stat Plot and determine which function best models the data

f(x) = x3 – 3x2 + x – 5

Determine Cubic Regression Equation and round coefficients to nearest integer STAT 6 ENTER

cubic


Waste Problem

The average daily amount of waste generated by each person in the United States is given below. This includes all wastes such as industrial wastes, demolition wastes, and sewage.

Yr 1980 1985 1990 1991 1992 1993 1994 1995 1996

lbs. of waste per

person per day

3.7 3.8 4.5 4.4 4.5 4.5 4.5 4.4 4.3

Is a linear function the best fit for this data?

6

5

4

3

6

5

4

3

f x = 0.05x+3.7

6

5

4

3

quadratic1 direction change

STAT 5 ENTER

Quadratic Regression

y = ax2 + bx + c a = -.004209b = .119480c = 3.592550R2 = .819795

y = -.004x2 + .119x + 3.593

6

5

4

3


Waste Problem

y = -.004x2 + .119x + 3.5936

5

4

3

6

5

4

3

a. Use the model to predict the amount of waste produced per day in 2010.

Since 2010 is 30 years later than 1980, find f(30).

f(30) = -.004x2 + .119x + 3.593 = 3.563 lb.

b. Use the model to predict when waste will drop to 3 pounds per day.

f(x) = -.004x2 + .119x + 3.593 = 3

x ≈ -4 or 34 1976 or 2014


6

4

2

Growth6

4

2

Decay

2

-2

-4

5

Growth2

-2

-5

Decay

Exponential Functions y = abx

Logarithmic Functions y = a + b ln x

Growth & Decay


Model Problem

The table below gives the population of the world in billions for selected years during the 1900’s.

YR

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

0 10 20 30 40 50 60 70 80 90 100

P 1.65 1.75 1.86 2.07 2.3 2.52 3.02 3.7 4.44 5.27 6.06

Determine an equation that models the data.


6

5

4

3

2

1

-1

20 40 60 80 100 120

Growth & Decay

0 10 20 30 40 50 60 70 80 90 100

P 1.65 1.75 1.86 2.07 2.3 2.52 3.02 3.7 4.44 5.27 6.06

y = ax + b a = .0435b = .97409r2 = .9008r = .9491y = .04x + 1

6

4

2

Growth

y = a · bx a = 1.4457b = 1.0136r2 = .9700r = .9849y = 1.44 ·1.01x

f x = 1.445691.01369x


4 pt. Regents Question

A biologist finds that a colony of bacteria grows exponentially and collects the following data on its size.

On a grid, make a scatter plot of this data. Write an exponential regression equation, expressing the regression coefficients to the nearest tenth.

Time(days)

Population (100s of liters per hour)

0 100

1 310

2 470

3 715

4 1150

5 1650

6 2500


Natural Log Growth Model Problem

The data in the table gives the yield y (in milligrams) of a chemical reaction after x minutes.x 1 2 3 4 5 6 7 8

y 1.5 7.4 10.2 13.4 15.8 16.3 18.2 18.3

Find a logarithmic model for the datay = 1.538+8.373 lnx


Oil Tanker Problem

An oil tanker collides with another ship and starts leaking oil. the Coast Guard measure the rate of flow of oil from the tanker and obtains the data shown in the table. Write a polynomial function to model the set of data.

Time(hours)

Flow Rate (100s of liters per hour)

1 18.0

2 20.5

3 21.3

4 21.1

5 19.9

6 17.8

7 15.9

8 11.3

9 7.8

10 3.7

f(x) = -0.4x2 + 2.8x + 16.3


Model Problem

Write a polynomial function to model the set of data.

x f(x)

-2.0 -22.0

-1.5 -7.9

-1.0 0.0

-0.5 3.5

0.0 4.1

0.5 2.9

1.0 2.1

1.5 2.5

2.0 5.8

2.5 14.1

3.0 28.0

f(x) = 2x3 – 3x2 – x + 4

aim: graph of best fit course: alg. 2 & trig. aim: how do we model real-world data with...

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